A New Method for Computing the Mean Center of Population
of the United States
Edward Aboufadel and David Austin
Grand Valley State University
The mean center of population of the United States is a convenient location used to summarize the population
distribution of the United States and how it changes over time. As computed by the U.S. Census Bureau, the
center depends on an arbitrary choice of a map projection. We feel that this location should depend only on the
population distribution and not on any choices made in representing the data of the distribution. This note
discusses a method for computing this location that does not depend on any choices made and describes how the
results of this method differ from those of the Census Bureau. Key Words: center of population, U.S. census,
U.S. Census Bureau.
n its decennial reports, the U.S. Census Bureau computes the mean center of population of the United States, a location designed to
summarize the nation’s population distribution
and its change over time. This location usually
receives considerable attention in the media (De
Jonge 2001; National Geographic 2002). The
method used by the Bureau to compute this
location first projects points from the threedimensional sphere to a flat map using a sinusoidal projection and then averages the twodimensional coordinates. Of course the choice
of a different map projection leads to a different
location for the center of population. We,
therefore, propose using a method that does
not depend on any choice of map projection. In
this note, we explain our method and compare it
to the Census Bureau’s method. We will see that
the center computed by the Census Bureau in
the 2000 Census lies roughly 125 km south of
the center computed with our method. This
discrepancy can be expected to increase, with
the Census Bureau’s center drifting further
south as the population becomes more spread
out across the country.
The general problem of defining a center of
population has been discussed by others, most
notably by Hayford (1902) and Bachi (1999).
The particular problems arising from the Census Bureau’s computation have been noted previously by Barmore (1993), who suggested
another method for computing the center of
population. Indeed, the bibliography of Bar-
I
more’s paper provides a good introduction to
this question. We describe and examine Barmore’s definition later in this article.
The Census Bureau has described its aim in
computing this center of population point:
The concept of the center of population as used
by the U.S. Census Bureau is that of a balance
point. That is, the center of population is the
point at which an imaginary, weightless, rigid,
and flat (no elevation effects) surface representation of the 48 conterminous states and the District of Columbia (or 50 states as appropriate to
the computation) would balance if weights of
identical size were placed on it so that each weight
represented the location on [sic] one person.
(U.S. Census Bureau 2001)
Defining the center of population as a balance
point is reasonable since the resulting location
has a simple intuitive meaning and may be easily
computed. Because the data points lie on Earth’s
curved surface, however, the meaning of the
word ‘‘flat’’ requires some interpretation. As we
will explain, the most unbiased interpretation is
one that does not rely on any choices made in
representing the data of the population distribution.
Beginning in 1960, the Census Bureau has
computed the center of population by dividing
the country into many small geographic areas
(eight million in the 2000 Census). For each of
these areas, the population is denoted by wi, the
latitude by fi, and the longitude by li. If the
center of population is the point with latitude f
The Professional Geographer, 58(1) 2006, pages 65–69 r Copyright 2006 by Association of American Geographers.
Initial submission, February 2004; revised submission, August 2004; final acceptance, December 2004.
Published by Blackwell Publishing, 350 Main Street, Malden, MA 02148, and 9600 Garsington Road, Oxford OX4 2DQ, U.K.
66
Volume 58, Number 1, February 2006
then the Bureau computes
and longitude l,
P
P
¼ Pi wi li cos fi :
¼ Pi wi fi l
ð1Þ
f
i wi
i wi cos fi
As Barmore (1993) has noted, this computation produces the point at which a flat map
would balance if the population of the United
States were placed on the map using a sinusoidal
(Sanson-Flamsteed) projection whose central
(Snyder
meridian is placed at longitude l.
[1987] has described many map projections, including those mentioned in this article.) One
notices that the latitude is simply averaged,
whereas the longitude is computed as an average
in which each person is weighted by the cosine
of his or her latitude, cos( fi). However, because
the range of latitudes spanned by the United
States is not large, this is not a significant factor:
and the average lonthe difference between l
P
P
gitude, computed simply as i wi li = i wi , is
approximately six minutes of longitude for the
2000 Census, a difference of only 8 km.
There appears to be no compelling reason for
choosing the sinusoidal projection over another
projection. Indeed, choosing different projections to represent the data leads to different locations for the center of population. Of course a
well-known consequence of Gauss’s (1828)
Thereoma Egregium is that it is not possible to
represent Earth’s surface on a flat map without
distorting distances (see Gauss’s theorem and a
discussion of its significance in O’Neill 1966).
Because a balance point depends on distances,
any computation that first projects the data onto
a flat map will not give the actual balance point.
For this reason, we propose considering this
computation from a three-dimensional perspective and finding the balance point of the
population distribution as it resides on an idealized spherical representation of Earth’s surface. As we will see, the result is a location that is
easily computed and does not depend on a
choice of map projection.
Following the Census Bureau, we divide the
country into regions whose population is wi and
whose location is represented by a three-dimensional vector xi extending from Earth’s center to the center of the region on Earth’s surface.
For convenience, we choose one unit of distance
to equal Earth’s radius and we follow the Census
Bureau in assuming that Earth is a perfect
sphere. To compute a balancing point, we recall
that the gravitational force on the population of
this region can be denoted as Fi ¼ kwixi, where
k is a constant that depends on the mass and
radius of the Earth.P
The total force
P on the population is then F ¼ i Fi ¼ k i wi xi , and the
balance point will be the location at which this
total force is directed toward Earth’s center.
This occurs at the location x where the total
force F is parallel to x, and consequently
F ¼ jFj
x, where |F| is the length of F. This
means that
P
F
wi xi
:
x¼
ð2Þ
¼ Pi
jF j
i wi xi
To compute this location, we first set up a threedimensional coordinate system on Earth. The
origin of this coordinate system is at Earth’s
center, the positive x-axis runs through the intersection of the prime meridian and the equator, the positive y-axis runs through the
intersection of the longitude at 901 East and
the equator, and the positive z-axis passes
through the North Pole. One unit of distance
is equal to Earth’s radius. In this way, a longitude-latitude pair (li, fi) gives the three-dimensional vector xi ¼ (xi, yi, zi) through the
relations
xi ¼ cos li cos fi
yi ¼ sin li cos fi
zi ¼ sin fi
To find the balance P
point, we then compute the
wi x i
balance point x ¼ Pi w x as the vector whose
j i i ij
coordinates are
P
P
P
w i xi
wy
wz
y ¼ P i i i z ¼ Pi i i :
x ¼ Pi
i wi xi
i wi x i
i wi xi
From here, we can recover the latitude and longitude of the center of population using
^ ¼ sin1 z
f
^ ¼ tan1 y
l
x
We note that the point given by Equation (2) is
also given by finding the weighted average of the
coordinates; that is,
P
P
P
w i xi
wi yi
wi zi
x^ ¼ Pi
; y^ ¼ Pi
; z^ ¼ Pi
:
w
w
i i
i i
i wi
From here, the balancing point is
x¼
ð^
x; y^; z^Þ
:
jð^
x; y^; z^Þj
A New Method for Computing the Mean Center of U.S. Population
Barmore (1993) discussed this approach to defining a center of population but rejected it because the point ð^
x; y^; z^Þ usually lies inside the
Earth (though x is a point on Earth’s surface).
We feel that the interpretation of x as a balancing point, together with the ease with which it
may be computed, make it the most appealing
candidate for the center of population.
To compare our method with that used by the
Census Bureau, let us first consider an unrealistic but simple test in which all of the population of the United States is concentrated in
equal numbers in Los Angeles (34N03,
118W15) and New York (40N43, 74W0). The
formulas used by the Census Bureau (1) give
¼ 37N23; l
¼ 97W07;
f
and those from the three-dimensional formula
(2) give
^ ¼ 37N31;
f
^ ¼ 97W10:
l
The difference is illustrated in Figure 1. The
two points computed are shown, as well as arcs
of great circles connecting them to New York
and Los Angeles. Notice that if we were to stand
at the point computed using the three-dimensional formula with a very light rod connecting
us to Los Angeles and another connecting us to
New York, the rods would be of equal length
and point in opposite directions. This is the
condition required for balancing. However, if
we were to stand at the point given by the formulas used by the Census Bureau (1), the two
rods would not point in opposite directions and
therefore would not balance.
It is important to note that the formulas used
by the Census Bureau pull the center of population south of its true location. These formulas
put the center of population roughly at the
midpoint of a segment drawn between the two
cities in the longitude-latitude coordinate
plane. However, great circles, the paths of
shortest distance and therefore the true
‘‘straight lines’’ on a sphere, travel a more
northerly route in the Northern hemisphere.
Turning now to the figures for the 2000 Census (see U.S. Census Bureau 2002), we find the
location given by the Census Bureau
¼ 38N47; l
¼ 91W34
f
67
Equation (2)
Census
Figure 1 A simple test case.
As shown in Figure 2, the two points differ by
some 125 km, with the point computed by the
Census Bureau, as expected, lying south of the
center given by Equation (2).
When viewed over the history of the Census,
the center of population gives an indication of
how the distribution of the population is changing. For instance, we certainly expect to see this
center move to the west as the population does.
For this reason, it is interesting to compute the
centers of population from 1790 through 2000
using Equation (2) and compare it to the center
reported by the Census Bureau. (Data for this
computation are given in U.S. Census Bureau
1993.) The results are shown in Figure 3. As
expected, both sets of points move to the west.
However, the point computed by the Census
veers south more quickly than the center computed using (2). Furthermore, the difference
between the two sets of points grows over time.
This is because the Census Bureau’s formulas
result from considering the population as if it
Corrected Center
100
Census Center
0
km
compared to our computation of
^ ¼ 38N42;
f
^l ¼ 91W49:
Figure 2 Centers of population for the 2000
Census shown in the Mercator projection.
Typos that were not fixed in the final version. It should read:
The Census Bureau reports 37N42, 91W49 and our
computation gives 38N47, 91W34.
68
Volume 58, Number 1, February 2006
Corrected
Census
2000
1950
1900
1850
1800
km
0
500
Figure 3 Centers of population, 1790–2000, shown in the Mercator projection.
were laid out in a flat plane. When the population was concentrated in a small area, this was
approximately true. However, as the population
has become more uniformly distributed across
the country, the effects of Earth’s curvature have
become more pronounced and the southerly
drift continues to grow.
The problem with the Census Bureau’s formulas was noted by Barmore (1993), who proposed another method for finding the center of
population. Barmore’s center of population is a
point that, when used as the center of an azimuthal equidistant projection, is a balance point
for the two-dimensional distribution. Barmore
chose this projection because such a map accurately represents the distance between the center and any other point. However, Barmore’s
definition suffers from a few drawbacks that
make it less desirable. First, it is unclear what the
intuitive meaning of this point is. A simple example shows that it is different from the balance
point. For instance, if we consider an idealized
example in which all the population is concentrated in New York and Los Angeles with New
York having twice the population of Los Angeles, Barmore’s center is located 12 miles west of
the balance point as found in (2) above. Second,
Barmore’s center must be computed through an
iterative approach that is technically cumbersome. Finally, it is relatively straightforward to
find examples in which this center is not unique.
In summary, the mean center of population is
meant as a convenient summary of the popula-
tion distribution. However, the method used by
the Census Bureau requires that an arbitrary
choice of map projection be used, and this
choice results in the center drifting south of a
less-biased computation. The method proposed
in this paper possesses some advantages in that it
has intuitive meaning as a balance point, is free
of any choices made in representing the data,
and is relatively simple to compute.’
Literature Cited
Bachi, R. 1999. New methods of geostatistical analysis and
graphical presentation. New York: Kluwer.
Barmore, F. E. 1993. Where are we? Comments on the
Concept of ‘‘Center of Population.’’ The Wisconsin
Geographer 9:8–21. (Also available in Solstice: An
Electronic Journal of Geography and Mathematics
1992 3 (2). Available at http://www-personal.
umich.edu/ copyright/image [last accessed 19
October 2005]).
De Jonge, P. 2001. The Zip in the middle. National
Geographic, November: 114–21.
Gauss, K. F. 1828. Disquisitiones generales circa superficies curvas [General investigations of curved surfaces]. Göttingen: Dieterich.
Hayford, J. F. 1902. What is the center of an area, or
the center of a population? Publications of the American Statistical Association 8 (58): 47–58.
National Geographic. 2002. A nation on the move. May.
O’Neill, B. 1966. Elementary differential geometry.
New York: Academic.
Snyder, J. P. 1987. Map projections—A working manual.
U.S. Geological Survey Professional Paper 1395.
A New Method for Computing the Mean Center of U.S. Population
Washington, DC: United States Government
Printing Office.
U.S. Census Bureau. 1993. Population: 1790 to 1990.
http://www.census.gov/population/censusdata/table-16.pdf (last accessed 8 December 2004).
———. 2001. Centers of Population Computation
for 1950, 1960, 1970, 1980, 1990 and 2000. http://
www.census.gov/geo/www/cenpop/calculate2k.pdf
(last accessed 8 December 2004).
———. 2002. Centers of Population for Census 2000.
http://www.census.gov/geo/www/cenpop/cntpop2k.
html (last accessed 8 December 2004).
69
EDWARD ABOUFADEL is a Professor in the
Department of Mathematics at Grand Valley State
University, Allendale, MI 49401. E-mail: aboufade
@gvsu.edu. His research interests include wavelets
and their application to areas such as object recognition.
DAVID AUSTIN is a Professor in the Department of
Mathematics at Grand Valley State University, Allendale, MI 49401. E-mail: austind@gvsu.edu. His professional interests include computing and the use of
computer graphics to communicate mathematics.